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Thursday, June 28, 2012

I'm traveling for 3 weeks, visiting family and (in 10 days) attending the Math Circle Teacher Training Institute. I'm not online much during my travels, and probably won't have much to say until I get back home. But I'm scanning through my stash on Google Reader, and have come across a few posts I'd like to share:

The story of some home-made stuffed Platonic solids (a gift for a baby) on Moebius Noodles might inspire me to try my hand at sewing some up. Beforehand, I'd want to figure out how to make them all about the same size. Should I go for the same height? Figuring that out would be a fun challenge.

Wednesday, June 20, 2012

Check it out. And 'like' it if you'd like to hear my frequent announcements about each current bit of progress on the book. I'll be setting up a newsletter soon that will have less frequent, more polished announcements about how the book is coming along.

Tuesday, June 19, 2012

The Bay Area Circle for Teachers is hosting a week-long workshop, and I got to do a math circle there this morning. I decided to do the Spot It analysis again. I had a group of 6 teachers, ranging from a 2nd/3rd grade teacher to high school teachers. They were all eager and persistent, and we'd did lots of good thinking together.

They played the game in pairs, and one pair started analyzing it before everyone was even done playing the game. I'd guess more than half the time was spent with participants working in pairs, with us working as one group together the rest of the time. I provided cards for them to make their own decks (half-size, 2"x3", fun colors, from Office Depot), which some of them used. Yesterday I was working on organizing those quotes from Bob that all involve keeping the discussion in the hands of the participants, so I was well-prepared when one person asked, "Does ___ work?", to reply, "That's an interesting question," and wrote her question on the board.

One person made a deck with 3 symbols - actually, we called them objects today - per card. A few made decks with 4 objects per card. We put a table on the board with 4 columns: # of cards, # of objects per card, # of objects total, total # of appearances of each object. We said a proper deck would have each card matching each other cards once, and found two different decks for 4 objects per card. Our smaller deck had 5 cards, 10 objects, each appearing twice. Our bigger deck had 13 cards, 13 objects, each appearing 4 times. We looked at whether our smaller deck was minimal (the smallest possible deck for cards with 4 objects per card) and whether our bigger deck was maximal. We wondered whether there were any proper decks with sizes between 5 and 13 cards (for 4 objects per card).

I recently heard the phrase Noticing and Wondering, and used it to start the Math Teachers at Play post. (Unfortunately, I don't remember where I heard it.) I'm loving it as a framework. Just two words. Just two simple questions: What do you notice? What do you wonder? I asked that today.

We had about 2 1/2 hours, and figured out some good relationships. We did not answer the question of 'How did they do that?' (How did the company that makes Spot It manage to create 55 cards, with 8 pictures on each one, with every pair of cards having exactly one match.) But we made some significant progress toward that goal, and it was exciting to see the thinking the participants did. I'd love to run a 6-week math circle starting from this game.

Our planet is shaped by the oceans, the dynamic geology and the changing
climate. It teems with life and we, in particular, have a massive
impact as we build homes, grow food, travel and feed our ever-hungry
need for energy. Mathematics is vital in understanding all of these,
which is why 2013 has been declared as the year for the Mathematics of Planet Earth.
As well as encouraging research into fundamental questions about the
Earth and how to meet the challenges it faces, there will also be many
opportunities during 2013 for everyone to get involved including public
lectures and workshops, competitions and exhibitions. The first such
competition is now underway: the MPE 2013 competition to design an exhibit about the mathematics of Planet Earth.

_____ [At some point, I decided I didn't like writing posts that just linked to someone else's post. I do that seldom now. That's why I had so many goodies to share in the Math Teachers at Play blog carnival. There were about 20 submissions and over 30 posts I had saved over the past year. But this one felt worth it.]

Is counting on your fingers good, bad, or both? What do you think? Peter (Classroom Professor) analyzes the mathematical thinking in two classrooms, (giving finger counting a thumbs down, and visualizing a thumbs up) and Caroline (Maths Insider) says don't count onyour fingers! (In a series debunking math myths, I said go ahead, but these posts got me thinking. Now I'd say it's a starting point, and let's think about how to move on.)

John (Math Hombre) has made a game where you multiply and divide by fractions to make the superheroes shrink and grow, Size the Day.

John Henry is the legend of a steel-driving man who competed against a steam engine. David (Delta Scape) shares this story with school children, and then they time each other doing multiplication worksheets with and without a calculator. "Students are asked to predict which method will take longer, gather data, and compare the results using box-plots." Sounds like a good way to help students decide when it makes sense to use a calculator.

Michelle (The Rookery) used Incredible Comparisons: The World in One Day(only available used) to help her students understand rates. I've visited Michelle's classroom, and it feels magical to me. Here's a quote from a post on playing class games, "A
child who has fallen on his knees to plead with another child to 'smile if you love me' does not feel inhibited when it's time to raise
his hand and take a guess at how to solve a math problem."

(from Christy's Game of Patterns)

Patterns & Logic

Making up your own games is super-engaging. Christy (Just Another Step...) and her son made up a game of patterns that they had great fun with.

Logic puzzles can be a great side door into the mansion of math. (Think about how Sudoku has swept the country.) The Island of Liars and Truthtellers is a classic setting for logic puzzles. Dan (mathrecreations) shares some background and 5 puzzles.

Dr. Techniko's game, How To Train Your Robot, sounds like a blast, suitable for very young kids, whose 'robots' are their parents.

(from britton.disted.camosun.bc.ca/jbsymteslk.htm)

Visual Math

(by Anna Weltman)

Becky (Wide Open Campus) shares photos from her son Z's Escheriffic day, along with a link to a tessellation maker and some thoughts on the magic.

In Not Just Shapes, Malke (The Map is Not...) continues her delightful series documenting her daughter's math discoveries. "As the structure of the universe continues to emerge in front of her
very own (and open) eyes, how much more fun will her world be to play
in, explore, put together, and then take apart again?"

Justin (Math Munch) shares Star Art with the readers of Math Munch (a weekly math links blog), along with some puzzle news. (Links to directions for making this beautiful blue star are in the comments.)

Rachel (Plus Magazine) wrote Shattering Crystal Symmetries. If I understand correctly, chemists used to think crystals were always organized in a repeating pattern; Dan Schectman analyzed the structure of a crystal that could not have a repeating pattern, and won the 2011 Nobel Prize for chemistry for this work, which is based on the mathematical work of Roger Penrose. Amazing! "Not only had mathematicians extensively studied symmetry, but, as
mathematicians are prone to do, they were also interested in how to
break it."

In Perspective in Math and Art, Annalisa (at Inside Higher Ed) writes about how learning to draw in perspective can be a bridge to learning math. "If you sketch a picture of the rails of the train track going into the
distance, and you know where the first two railroad ties go, where do
you put the next one?"

(from Fawn's area of a circle lesson)

Algebra, Geometry, &Trigonometry

Kids are never too young to do some algebraic thinking with the Function Machine or Guess My Rule game. Denise (Let's Play Math!) spells it out carefully, and John (Math Hombre) writes about using it with college students, "7 to 1 and then 1 to 7 drew an audible gasp."

Smruti (Maths Study Blog) shows a method for finding simple side lengths when you have one side of a right triangle. One side is not enough to establish just one possible triangle, but if you'd like to play with finding Pythagorean triples (3 whole numbers giving the lengths of sides of a right triangle, like 3-4-5), then this technique is intriguing. [His site has flashing ads and brought up a pop-up window. I believe it's safe, but can't be sure.]

Haggis (Knot Your Average Sheep) helped design some activities for an interactive evening at the museum (National Museum of Scotland), and included this: "Can you colour the lines [on the star above] with 3 colours so that at each star 3 different colours meet?"

RobertAbbott (inventor of the card game Eleusis) has shared some great online Logic mazes.

Here's a puzzle from Alexander (Cut the Knot): Given a sequence of numbers, pick any two, say A and B, randomly and replace the two with the result of A×B+A+B. Repeat the procedure until only one number remains. Try to predict the final result. You can play with it online. What's happening?

(from Rick's blog banner)

Notation and Language

Sometimes the notation makes a math topic harder than it needs to be. Take logarithms, for example. Where'd that word come from, anyway? Kate (f(t)) uses power2(8) = 3 to invite her students to figure out what the new function is. I used her idea, but changed it to P2(x); it worked great.

Rick (Exploring Binary) wanted a word for the portion of a binary number after the ... umm, "decimal" point. You know, the part that represents a fraction. He wants to know if 'bicimal' works for you.

Peter and Christian (The Aperiodical, a math links blog) found a CNN story on an advance that may change public transportation, based on linear algebra. If a bus will be running more often than every 10 minutes, passengers can wait less if there's not a schedule. Each bus stops at each end of the line the right amount of time to average its time between the bus in front and behind it. Bus bunching (where one bus ends up right behind another) has always been a big problem, and this solves it. Most of the mathematical paper is quite readable.

Denise (Let's Play Math!) says, "What better way could there be to do math than snuggled up on a couch
with your little one, or side by side at the sink while your
middle-school student helps you wash the dishes, or passing the time on a
car ride into town?" Mmm, tell me a math story, please.

If you want students to learn math through projects (Project-Based Learning has its own acronym of course, PBL), you need to come up with projects that fit your subject and your teaching style. Bryan (Doing Mathematics) brainstorms some enticing ones. Geoff (emergent math) makes a plea for more inquiry-based lessons (is that the same as project-based?) He has set up a google docs repository for each course from algebra to calculus, and lots of folks have contributed ideas. You can use theirs or add some more.

And now we've come to the end of the 51st Math Teachers at Play blog carnival. Here's one last parting thought... I once read that, among the Tsilagi (Cherokee), you become an adult at 51. (Perhaps that's a bad translation, and you become an elder at 51?) That idea really stuck with me, and when I turned 51 I thought often about how much I'd matured since I was 18. With a 10-year-old in my life, I'm still working hard at maturity... What's 51 mean to you?

The next Math Teachers at Play blog carnival will be hosted at Denise's Let Play Math! blog in the 2nd week of July. If you'd like to be a host of this monthly carnival, check here for open dates. Until then, take your time to savor all these goodies, and when you're done here, check out the 87th Carnival of Mathematics at Random Walks.

Monday, June 11, 2012

Nope, I won't be on the radio; it's for their website. I was asked questions for an article about "helping children develop a good attitude toward math". I'll let you all know when it's published. (I'm very excited. I hope she mentions our book, Playing With Math: Stories from Math Circle, Homeschoolers, and Passionate Teachers.)

Sunday, June 10, 2012

If someone says they 'understand' something, what does that mean? Elementary teachers often think explaining why means giving a cute rhyming 'reason',* a shocking thought to me. And a stark reminder that we each have our own ideas about the meanings of the most basic words.

Stieg Mellin-Olsen**, of Bergen University, suggests that there are in current
use two meanings of the word 'understanding'. These he distinguishes by calling them
‘relational understanding’ and ‘instrumental understanding’. By the
former is meant what I have always meant by understanding, and probably
most readers of this article: knowing both what to do and why.
Instrumental understanding I would until recently not have regarded as
understanding at all. It is what I have in the past described as ‘rules
without reasons’, without realising that for many pupils andtheir teachers the possession of such a rule, and ability to use it, was what they meant by ‘understanding’.

Suppose that a teacher reminds a class that the area of a rectangle is given by A=L×B. A
pupil who has been away says he does not understand, so the teacher
gives him an explanation along these lines. “The formula tells you that
to get the area of a rectangle, you multiply the length by the breadth.”
“Oh, I see,” says the child, and gets on with the exercise. If we were
now to say to him (in effect) “You may think you understand, but you
don’t really,” he would not agree. “Of course I do. Look; I’ve got all
these answers right.” Nor would he be pleased at our devaluing of his
achievement. And with his meaning of the word, he does understand.

No, there is no way around it. If you want students to have meaningful
learning experiences that culminate in transferable insight and
know-how, then you have to lose time to gain it. You have to slow down
the teaching to speed up the learning.

_____* In the experience of a wise friend who works with many elementary teachers, trying to help them improve their mathematical understandings.** Also of interest, The Politics of Mathematics Education, by Mellin-Olsen.

Saturday, June 9, 2012

Dan Goldner at Work in Pencil, just wrote a great post sharing a student's response to his course. He linked to the Journal of Inquiry-Based Learning in Mathematics, which I hadn't seen before. Interestingly, it doesn't seem to have 'issues' like most journals. Instead, it's a repository of refereed course notes for college math courses. Dan pointed to the Trigonometry notes. There are also notes for Probability and Statistics, Calculus, and Linear Algebra. I think the rest are all above the level I teach (community college), but I might want to play with the geometry and/or number theory notes.

Friday, June 8, 2012

Last weekend I went to a workshop on teaching and learning Latin called Where are Your Keys (WAYK). I had never taken a Latin class in my life, and I didn’t go to the workshop because of a sudden interest in Latin. I went because WAYK is a philosophy, a community, and a bunch of techniques for learning. Although WAYK is specifically for learning languages, I think that much of the philosophy and many of the techniques can be adapted to make learning anything more fun and more efficient.

I wish I could remember how I heard about them. It was in mid or late April, and I was pretty excited when I stumbled on the WAYK site. I've been thinking about buying a Rosetta Stone language course (stopped by the high price). My son is most interested in Hebrew, and I'm open to learning any new language. I might have been exploring that sort of thing. I got on their email list, and found out about a weekend workshop in Santa Rosa. I was already booked that weekend, but I managed to squeeze in two trips up there, on Friday evening and Sunday morning.

I loved it, and met a young man there who is very involved with WAYK and loves math. He was wishing he'd had access to their techniques in his Calc II class. I took that as a good sign, and on Sunday night I started to make a handout to explain this to my students. I'm not sure yet how to modify it all for math class, but I'm hoping to test out some of my ideas this summer. (If you'd be willing to meet to try something out, let me know.) I thought I'd post what I have so far, in case anyone else is excited by this.

When I read John Spencer's post on the problems with gamification, I wanted to show this to him. I don't like gamification much either, with its emphasis on external rewards and its notion that video games are the games to imitate. I hadn't even thought about how WAYK is related to all that until I read John's post. WAYK treats learning as a game, but in a very different way than the gamification I've heard about. It's not about points, badges, or quantifying what you've learned, and it's not rigid. It does encourage 'small bites' (John's issue #3 with gamification).

Evan Gardner put together WAYK to help a native community with its language revitalization project. He used lots of techniques already used in language courses, combined with turning it all into a big game, and using sign. You can watch this video to get a sense of how it works.

The use of sign helps the learning process in a number of ways:

Since sign language uses a different part of the brain than spoken
languages, it interferes less with learning the new spoken language. (In the case of math, I'm not sure how this benefit translates.)

Moving our bodies and associating each idea with a movement both help us retain our new learning.

Communicating your needs with sign doesn't interrupt
the flow.

The name 'where are your keys?' comes from the game's dependence on
props. Keys are something almost any adult you talk to will have in
their pocket. Great for learning language through conversation. Nothing
to do with math. So I'm naming my translation attempt 'The Learning Game'.

Throughout the workshop, I was thinking about how each of the techniques would translate to math class. Some translate easily and some might not work at all. What I'm including below is a rough first pass. Some of it will be on a handout I give to students; some won't. The game has lots of techniques, too many to learn at first. (In fact one technique, 'small bite,' probably explains why Evan doesn't have lists of the techniques. It's too much at once.)

Philosophy & First Principles

The game: Learning is something our brains love. (Unless they’ve been traumatized by schools and tests.) We want to turn the whole process into a game. Some games are about who wins. This game is for everyone to win. Peek-a-boo is a game you play with babies. Everyone wins. Hide-and-seek is not about winning – it’s about playing. Think of this game as a treasure hunt or a scavenger hunt, where everyone who finds the treasures' hiding places gets treasure. We will play the game best (most fun and most effectively) if we turn this class into a community where each person figures out how to play the game in a way that helps them and those around them to learn efficiently (this means deeply understanding the ideas, connecting them, using them, creating your own, and communicating them).

The techniques: All games have rules. The rules of this game will be called techniques. Each technique is meant to be a ‘learning accelerator/deepener’. You might figure out a new technique (especially now, because the game is pretty new). If you do, please introduce it to the group / class / community - this game is meant to be modified.

Meta-Learning: Although you are officially learning _[algebra, calculus,...]_ in this class, any class is a chance to learn how to learn. What you learn from this game may be more powerful than the ______ you learn. Or, it may help you learn ____ much more deeply than you ever could have without it.

Signing: Each technique has a sign associated with it. We’ll also invent a sign for each concept we want to learn. Signing allows us to use a different part of our brain for the game, and it associates the new ideas with a physical movement, which deepens our learning. (Just moving helps the learning too.) To see how this works, watch this.

Teach it: When there’s one teacher and x students (with x>10), we’re tempted to revert to teacher talking, students listening. But that leaves students too passive, which is not effective for learning anything. Nowadays there are fine ways to offer information to every student in the world through the internet; a class needs to do something better. We’re a learning community, playing together. We'll split into groups (of 3 to 5 people each) as much as possible, with students teaching each other. You’ll know you’ve learned it when you can teach it. Trying to teach it will show you what you need to learn.

Techniques (aka Rules of the Game)
Remember, it's all about taking over the learning process, as a community. The leader knows a lot about math and about learning, but only you know what's going on for yourself at each moment. Use the techniques to make this learning community work for you.

There are too many techniques here to start out with. If you could only introduce 5 on the first day, which would you choose?

Changing the Pace
You can use / ask for:

Technique: Slow when you want someone to slow down (sign: pull hand slowly up other arm)

Technique: Again when you want something repeated (sign: tap end of fingers into cupped other hand)

Technique: No pressure refresher to ask for a recap

Technique: Need help when you're feeling lost

Technique: Meet me (where I am) when the level is too high (sign: I think it will be a modified meet sign (both hands have index finger pointing up), with one hand higher at first, pulled down to meet the other. I'd like to get Evan's advice on this.)

[Note: WAYK calls this last one "Sorry Charlie," in reference to their name for the levels of proficiency, which is 'Travels with Charlie'. I didn't want to use a male name in this, so I've changed the names of both these techniques. I also wanted to emphasize that it's ok to be at any level. I wanted the name of this technique to have a zen feel to it. We are where we are, and we take our first step from there. In math classes, people have lots of baggage about being ‘behind’, not wanting to slow the class down. I want this technique to help them past that.]

Technique: How fascinating! We need to celebrate our mistakes. Noticing what we did wrong, and enjoying the process of seeing how that’s different from what really works is our goal. Also great to do this when you have a sudden insight, or love what someone else just said. (sign: arms high, wiggle hands) [I love that this celebrates both right and wrong answers.]

Technique: Show your level: Proficiency levels: In a language, you have an overall proficiency. In math, you have proficiencies with different topics. (sign: one hand held out with the four fingers spread out, other hand points to where you are on the scale)We'll start with this:

I think I get (much of) what you’re saying, not ready to do anything with it yet.

I’m ready to try the simplest case, if I can work with a partner.

I can do this.

I’m ready to teach it to my group.

[I've been using 'thumbs up-down-sideways' for years now. It will be hard to switch, but I think this will be more useful. Since I've modified this, it hasn't been tested. My students can help me improve it.

The WAYK version is all about communication:

Tarzan at the party (or Sesame Street)

Where is the party? (or Dora the Explorer and Mr. Rogers)

What Happened at the party last night? (or Larry King and Oprah)

What if parties were illegal? (or Charlie Rose)

Charlie Rose gave this scale its 'Travels with Charlie' name, and the 'meet me where I am' technique its 'Sorry Charlie' name.]

Technique: Three times: Anything new is repeated 3 times. In
math, it’s often one time where I do all the steps, a second time where I
ask students to walk through the steps more with me, and a 3rd time
where students do it in pairs. A student who wants to succeed will do
the exercises again at home, where they do each of those 3 problems
without looking at the solution, and then do similar problems on their
own. Maybe that’s 3x3.

Technique: Full: Show how full you are. Be aware of when your brain needs a break. The idea here is that we take care of ourselves and each other. If many people are too full, perhaps we need a 'no pressure refresher'.

Technique: Mumble: It’s ok to do something that doesn’t quite make sense at first

Technique: Sidekick: If you're leading, you can get help from a friend.

Technique: Bucket brigade: Get the new stuff from the elder / ‘teacher’ and pass it along in buckets (groups).

Technique: My turn / your turn: If you’re the one doing the routine, use this sign to hand the floor over to someone else, who then repeats the process. Not sure how we’ll use this one for math, but I want to throw it out there in case we find it useful.

Technique: Set up: Finding props ahead of time that will make the distinctions clearer. For math, this may mean finding cool problems.

Technique: Limit: Limit how much you deal with at on time, akaTechnique: Bite-size piece

Technique: Hunting: We’re trying to figure out if we understand, so we hunt down something we can try to test our understanding. This is a way to take your learning into your own hands.Technique: Prove it: Apply this bite sized piece to something new

Technique: Clear the field: If the board has too much on it, this is a request to erase the board

Technique: Let’s do it: To ask the leader to turn it over to groups to practice.Technique: Share the wealth: Teach another what you know

Technique: Distraction: (sign: wiggle hand near side of head, point either to someone whose help you’d like or to the person who is distracting you) because it’s a game, it loses the tatlletale sting.

Can I do it?
This is an awful lot of not-math to be introducing to students who have an image of how math class works imprinted on their brains. But those images get us in trouble. For instance, students think "The teacher will tell me the right answer. " (Nope, says Kate.)

I'm hoping I can get them working on a math problem the first day, and introduce this alongside it. I don't have it worked out yet...

What do you think? Am I crazy to load all this extra stuff onto a math class? Can it work? Want to play The Learning Game with me sometime this summer?

Thursday, June 7, 2012

I’m teaching a summer physics course. One thing I’m doing differently
this year is having students do at-home experiments with friends or
family. Part of their reporting back involves having to share the ideas
of their friend or family member. Here are a few quotes from students
discussing what happened when they dropped a book, a piece of paper, and
crumpled up piece of paper.

I think the most interesting thing for the physics teacher is people's explanations of why things happened.

I'm trying to think of topics in my math classes that students could share with friends and family. I'll be teaching pre-calculus, calculus I, and calculus II in the fall. But I'm interested in algebra questions that can be shared at home, too. The first criteria is ease of understanding, but there's also the aha! factor, when something doesn't turn out like you expected. I know lots of cool mathy experiments that students could easily share, like making a mobius strip, but I'm not sure I know any related to calculus...

Tuesday, June 5, 2012

... Math Teachers at Play #51 will be coming to my blog next week. I need your help. Send me your favorite playful post on math (little kids up through calculus level), new or old (as long as it hasn't been in the carnival before). I'd like to receive it by this Friday.*